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This section describes the discretization schemes of convective terms in finite-volume equations. The accuracy, numerical stability, and boundness of the solution depend on the numerical scheme used for these terms. The central issue is the specification of an appropriate relationship between the convected variable, stored at the cell center, and its value at each of the cell faces.

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Here is described the discretization schemes of the convective terms in the finite-volume equations. The accuracy, numerical stability and the boundness of the solution depends on the numerical scheme used for these terms. The central issue is the specification of an appropriate relationship between the convected variable, stored at the cell centre and its value at each of the cell faces.

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== Basic Equations of CFD ==

== Basic Equations of CFD ==

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</td><td width="5%">(1)</td></tr></table>

</td><td width="5%">(1)</td></tr></table>

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[[Image:Stencil_3a.jpg]]

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Equation (1) is integrated over a control volume and the following discretized equation for <math>\boldsymbol{\phi}</math> is produced:

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Equation (1) is integrated over a control volume and the following discretised equation for <math>\boldsymbol{\phi}</math> is produced:

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<table width="100%"><tr><td>

<table width="100%"><tr><td>

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</td><td width="5%">(1)</td></tr></table>

</td><td width="5%">(1)</td></tr></table>

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where <math>\boldsymbol{C_{f}}</math> is the mass flow rate across the cell face <math>\boldsymbol{f}</math>. The convected variable <math>\boldsymbol{\phi_{f}}</math> associated with this mass flow rate is usually stored at the cell centres, and thus some form of interpolation assumption must be made in order to determine its value at each cell face. The interpolation procedure employed for this operation is the subject of the various schemes proposed in the literature and the accuracy, stability and boundedness of the solution depends on the procedure used.

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where <math>\boldsymbol{C_{f}}</math> is the mass flow rate across the cell face <math>\boldsymbol{f}</math>. The convected variable <math>\boldsymbol{\phi_{f}}</math> associated with this mass flow rate is usually stored at the cell centers, and thus some form of interpolation assumption must be made in order to determine its value at each cell face. The interpolation procedure employed for this operation is the subject of the various schemes proposed in the literature, and the accuracy, stability, and boundedness of the solution depend on the procedure used.

In general, the value of <math>\boldsymbol{\phi_{f}}</math> can be explicity formulated in terms of its neighbouring nodal values by a functional relationship of the form:

In general, the value of <math>\boldsymbol{\phi_{f}}</math> can be explicity formulated in terms of its neighbouring nodal values by a functional relationship of the form:

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where <math>\boldsymbol{\phi_{nb}}</math> denotes the neighbouring-node <math>\boldsymbol{\phi}</math>values.

where <math>\boldsymbol{\phi_{nb}}</math> denotes the neighbouring-node <math>\boldsymbol{\phi}</math>values.

All the convection schemes involve a stencil of cells in which the values of <math>\boldsymbol{\phi}</math> will be used to construct the face value <math>\boldsymbol{\phi_{f}}</math>

All the convection schemes involve a stencil of cells in which the values of <math>\boldsymbol{\phi}</math> will be used to construct the face value <math>\boldsymbol{\phi_{f}}</math>

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[[Image:NM_convectionschemes_Stencil_2a.jpg]]

Where flow is from left to right, and <math>\boldsymbol{f}</math> is the face in question.

Where flow is from left to right, and <math>\boldsymbol{f}</math> is the face in question.

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<math>\boldsymbol{D}</math> - mean '''D'''ownstream node

<math>\boldsymbol{D}</math> - mean '''D'''ownstream node

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In the first plot, it is not so natral to think the central node "C" not as the present node "P". It may be thought as the first node to the upstream direction of the surface in question "f".

== Basic Discretisation schemes ==

== Basic Discretisation schemes ==

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This scheme is 2nd-order accurate, but is unbounded so that non-physical oscillations appear in regions of strong convection, and also in the presence of discontinuities such as shocks. The CDS may be used directly in very low Reynolds-number flows where diffusive effects dominate over convection.

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[[Image:NM convectionschemes CDS 02.jpg]]

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This scheme is 2nd-order accurate, but is unbounded so that unphysical oscillations appear in regions of strong convection and also in the presence of discontinuities such as shocks. The CDS may be used directly in very low Reynolds-number flows where diffusive effects dominate over convection.

HRS can be classified as ''linear'' or ''non-linear'', where ''linear'' means their coefficients are not direct functions of the convected variable when applied to a linear convection equation. It is important to recognise that linear convection schemes of 2nd-order accuracy or higher may suffer from unboudedness, and are not unconditionally stable.

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HRS can be classified as ''linear'' or ''non-linear'', where ''linear'' means their coefficients are not direct functions of the convected variable when applied to a linear convection equation. It is important to recognise that linear convection schemes of 2nd-order accuracy or higher may suffer from unboundedness, and are not unconditionally stable.

''Non-linear'' schemes analyse the solution within the stencil and adapt the discretisation to avoid any unwanted behavior, such as unboundedness (see Waterson [1994]). These two types of schemes may be presented in a unified way by use of the ''Flux-Limiter'' formulation (Waterson and Deconinck [1995]), which calculates the cell-face value of the convected variable from:

''Non-linear'' schemes analyse the solution within the stencil and adapt the discretisation to avoid any unwanted behavior, such as unboundedness (see Waterson [1994]). These two types of schemes may be presented in a unified way by use of the ''Flux-Limiter'' formulation (Waterson and Deconinck [1995]), which calculates the cell-face value of the convected variable from:

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The generalisation of this approach to handle non-uniform meshes has been given by Waterson [1994]

The generalisation of this approach to handle non-uniform meshes has been given by Waterson [1994]

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From equation (\ref{eq9}) it can be seen that <math>\boldsymbol{\varphi=1}</math> gives the UDS and <math>\boldsymbol{\varphi=r}</math> gives the CDS.

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From the equation (\ref{eq9}) it can be seen that <math>\boldsymbol{\varphi=1}</math> gives the UDS and <math>\boldsymbol{\varphi=r}</math> gives the CDS.

The HRS schemes can be introduced into equation (\ref{eq4b}) by using the deffered correction procedure of Rubin and Khosla [1982]. This procedure express the cell-face value <math>\boldsymbol{\phi_{f}}</math> by:

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The HRS schemes can be introduced into equation (\ref{eq4b}) by using the deferred correction procedure of Rubin and Khosla [1982]. This procedure expresses the cell-face value <math>\boldsymbol{\phi_{f}}</math> by:

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This treatment leads to a diagonally dominant coefficient matrix since it is formed using the UDS.

This treatment leads to a diagonally dominant coefficient matrix since it is formed using the UDS.

*'' Passing through <math>\boldsymbol{Q}</math> with a slope of 0.75 (for a uniform grid) is necessary and sufficient for third-order accuracy''

*'' Passing through <math>\boldsymbol{Q}</math> with a slope of 0.75 (for a uniform grid) is necessary and sufficient for third-order accuracy''

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The horizontal and vertical coordinates of point <math>\boldsymbol{Q}</math> in the normalized variable diagram and the slope of the characteristics at the point <math>\boldsymbol{Q}</math> for preserving the third-order accuracy for a nonuniform grid can be obtained by simple algebra using eqs. [.....]

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The horizontal and vertical coordinates of point <math>\boldsymbol{Q}</math> in the normalized variable diagram, and the slope of the characteristics at the point <math>\boldsymbol{Q}</math> for preserving the third-order accuracy for a nonuniform grid, can be obtained by simple algebra using eqs. [.....]

The CBC is clearly illustrated in figure below, where the line <math>\hat{\phi_{f}} = \hat{\phi_{C}}</math> and the shaded area are the region over which the CBC is valid. The importance of the CBC is to provide a sufficient and necessary condition for guaranteeing the bounded solution if at most three neighbouring nodal values are used to approximate face values. It is well known that the positivity of finite-difference coefficients is also a sufficient condition for boundedness, but this is overly stringent, for the existense of negative coefficients does not neccesarily lead to over- or undershoots.

The CBC is clearly illustrated in figure below, where the line <math>\hat{\phi_{f}} = \hat{\phi_{C}}</math> and the shaded area are the region over which the CBC is valid. The importance of the CBC is to provide a sufficient and necessary condition for guaranteeing the bounded solution if at most three neighbouring nodal values are used to approximate face values. It is well known that the positivity of finite-difference coefficients is also a sufficient condition for boundedness, but this is overly stringent, for the existense of negative coefficients does not neccesarily lead to over- or undershoots.

Introduction

This section describes the discretization schemes of convective terms in finite-volume equations. The accuracy, numerical stability, and boundness of the solution depend on the numerical scheme used for these terms. The central issue is the specification of an appropriate relationship between the convected variable, stored at the cell center, and its value at each of the cell faces.

Basic Equations of CFD

All the conservation equations can be written in the same generic differential form:

(1)

Equation (1) is integrated over a control volume and the following discretized equation for is produced:

(2)

where is the source term for the control volume , and and represent, respectively, the convective and diffusive fluxes of across the control-volume face

The convective fluxes through the cell faces are calculated as:

(1)

where is the mass flow rate across the cell face . The convected variable associated with this mass flow rate is usually stored at the cell centers, and thus some form of interpolation assumption must be made in order to determine its value at each cell face. The interpolation procedure employed for this operation is the subject of the various schemes proposed in the literature, and the accuracy, stability, and boundedness of the solution depend on the procedure used.

In general, the value of can be explicity formulated in terms of its neighbouring nodal values by a functional relationship of the form:

Convection Schemes

All the convection schemes involve a stencil of cells in which the values of will be used to construct the face value

Where flow is from left to right, and is the face in question.

- mean Upstream node

- mean Central node

- mean Downstream node

In the first plot, it is not so natral to think the central node "C" not as the present node "P". It may be thought as the first node to the upstream direction of the surface in question "f".

Basic Discretisation schemes

Central Differencing Scheme (CDS)

It also can be considered as linear interpolation.

The most natural assumption for the cell-face value of the convected variable would appear to be the CDS, which calculates the cell-face value from:

(1)

or for more common case:

(1)

where the linear interpolation factor is definied as:

(1)

normalized variables - uniform grids

(1)

normalized variables - non-uniform grids

(1)

This scheme is 2nd-order accurate, but is unbounded so that non-physical oscillations appear in regions of strong convection, and also in the presence of discontinuities such as shocks. The CDS may be used directly in very low Reynolds-number flows where diffusive effects dominate over convection.

Upwind Differencing Scheme (UDS) also (First-Order Upwind - FOU)

The UDS assumes that the convected variable at the cell face is the same as the upwind cell-centre value:

(1)

normalised variables

(1)

The UDS is unconditionally bounded and highly stable, but as noted earlier it is only 1st-order accurate in terms of truncation error and may produce severe numerical diffusion. The scheme is therefore highly diffusive when the flow direction is skewed relative to the grid lines.

(1)

(1)

UDS may be written as

(1)

or in more general form

(1)

where

(1)

(1)

Hybrid Differencing Scheme (HDS also HYBRID)

The HDS of Spalding [1972] switches the discretization of the convection terms between CDS and UDS according to the local cell Peclet number as follows:

(1)

(1)

The cell Peclet number is defined as:

(1)

in which and are respectively, the cell-face area and physical diffusion coefficient. When
,CDS calculations tends to become unstable so that theHDS reverts to the UDS. Physical diffusion is ignored when .

The HDS scheme is marginally more accurate than the UDS, because the 2nd-order CDS will be used in regions of low Peclet number.

Power-Law Scheme (also Exponential scheme or PLDS )

High Resolution Schemes (HRS)

Classification of High Resolution Schemes

HRS can be classified as linear or non-linear, where linear means their coefficients are not direct functions of the convected variable when applied to a linear convection equation. It is important to recognise that linear convection schemes of 2nd-order accuracy or higher may suffer from unboundedness, and are not unconditionally stable.

Non-linear schemes analyse the solution within the stencil and adapt the discretisation to avoid any unwanted behavior, such as unboundedness (see Waterson [1994]). These two types of schemes may be presented in a unified way by use of the Flux-Limiter formulation (Waterson and Deconinck [1995]), which calculates the cell-face value of the convected variable from:

(1)

where is termed a limiter function and the gradient ration is defined as:

(1)

The generalisation of this approach to handle non-uniform meshes has been given by Waterson [1994]

From the equation (\ref{eq9}) it can be seen that gives the UDS and gives the CDS.

Normalised Variables Formulation (NVF)

Normalised Variables Diagram (NVD)

According to Leonard [1988], for any (in general nonlinear) characteristics in the normalized variable diagram (see figure below):

Passing through is necessary and sufficient for second-order accuracy

Passing through with a slope of 0.75 (for a uniform grid) is necessary and sufficient for third-order accuracy

The horizontal and vertical coordinates of point in the normalized variable diagram, and the slope of the characteristics at the point for preserving the third-order accuracy for a nonuniform grid, can be obtained by simple algebra using eqs. [.....]

Gaskel and Lau have formulated the CBC as follows. A numerical approximation to is bounded if:

for , is bounded below by the function and above by unity and passes through the points (0,0) and (1,1)

for or , is equal to

The CBC is clearly illustrated in figure below, where the line and the shaded area are the region over which the CBC is valid. The importance of the CBC is to provide a sufficient and necessary condition for guaranteeing the bounded solution if at most three neighbouring nodal values are used to approximate face values. It is well known that the positivity of finite-difference coefficients is also a sufficient condition for boundedness, but this is overly stringent, for the existense of negative coefficients does not neccesarily lead to over- or undershoots.